The equations of motion are a long complicated set of coupled non-linear differential equations which require a numerical solution. However, before presenting the numerical results, it is instructive to extract some useful analytical information from the equations. In particular, we will determine the analytical behaviour of the fields near the origin and in the asymptotic region. We will see that the asymptotic profile of a vortex is not supported by a positive cosmological constant \(\Lambda _{eff}\) ; it must be either negative (AdS\(_3\)  background) or zero (Minkowski background). This is similar to the fact that in \(2+1\)  dimensional General Relativity (GR), a black hole exists for negative cosmological constant (the BTZ black hole [1]}, [2]}) but not for positive cosmological constant. There is no black hole in a Minkowski background either but in contrast, one can have a vortex in a Minkowski background.
